Curve detail
Definition
| Name | w-255-mers |
|---|---|
| Category | nums |
| Description | Original nums curve from https://eprint.iacr.org/2014/130.pdf |
| Field | Prime (0x7ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffd03) |
| Field bits | 255 |
| Form | Weierstrass $y^2 = x^3 + ax + b$ |
| Param $a$ | 0x7ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffd00 |
| Param $b$ | -0x51bd |
Characteristics
| Order | 0x7fffffffffffffffffffffffffffffff864a38283ad2b3dfab8fac983c594aeb |
| Cofactor | 0x1 |
| $j$-invariant | 0x661734f09b8540109c7681cdde324ef2080bff3ca6ef898c0002ffb7060110ba |
| Trace $t$ | 0x79b5c7d7c52d4c2054705367c3a6b219 |
| Embedding degree $k$ | 0x3fffffffffffffffffffffffffffffffc3251c141d6959efd5c7d64c1e2ca575 |
| CM discriminant | -0x1c622a801d47d31929427898fe6049161ccb6472bd76d9816a4f91a0475ad2d9b |
Traits
| $\text{cofactor}()$ | |
|---|---|
| order | 0x7fffffffffffffffffffffffffffffff864a38283ad2b3dfab8fac983c594aeb |
| cofactor | 0x1 |
| $\text{discriminant}()$ | |
| cm_disc | None |
| factorization | None |
| max_conductor | None |
| $\text{twist_order}(deg=1)$ | |
| twist_cardinality | 0x8000000000000000000000000000000079b5c7d7c52d4c2054705367c3a6af1d |
| factorization | None |
| $\text{twist_order}(deg=2)$ | |
| twist_cardinality | 0x3ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffd0239dd57fe2b82ce6d6bd8767019fb6e9e3349b8d4289267e95b06e5fb8a5bba75 |
| factorization | None |
| $\text{kn_factorization}(k=1)$ | |
| (+)factorization | ['0x2', '0x2', '0xd', '0x3b', '0x1fe1', '0xb48b', '0x4a38bf89ed42039', '0x1a3780881d4886ea3ba35bc58d18547c2ff8cac7'] |
| (+)largest_factor_bitlen | 0x9d |
| (-)factorization | ['0x2', '0x3', '0x3', '0x43', '0x9d', '0xb3', '0x2479', '0x681173', '0x44604a62c855bfe5e408585d7aee72b7d256d5ae08e72489b'] |
| (-)largest_factor_bitlen | 0xc3 |
| $\text{kn_factorization}(k=2)$ | |
| (+)factorization | NO DATA (timed out) |
| (+)largest_factor_bitlen | NO DATA (timed out) |
| (-)factorization | ['0x17b', '0x58bf34df5b2371', '0x1f2cd1b9937efdfba0ee166128f33469a232a57eb44e5965f'] |
| (-)largest_factor_bitlen | 0xc1 |
| $\text{kn_factorization}(k=3)$ | |
| (+)factorization | ['0x2', '0x47', '0x655', '0x1f14708cb', '0x498b7072d1', '0xc3ea7753e88ef71396824c0a8b0443303c1dd6c101'] |
| (+)largest_factor_bitlen | 0xa8 |
| (-)factorization | ['0x2', '0x2', '0x2', '0x2', '0x2', '0x2', '0x5', '0x5', '0x17', '0x8b', '0x13d', '0xb109b', '0xaaf7d9aa7', '0x1abb157f73b', '0x206b55777b741d', '0x28a915bd80a388c5f5f1'] |
| (-)largest_factor_bitlen | 0x4e |
| $\text{kn_factorization}(k=4)$ | |
| (+)factorization | ['0xb', '0x11', '0x4577', '0x1bc13', '0x7fce5', '0x10d8b79b09d', '0x4074630080e7b00528ed', '0x2bf386b39ad63fe35bd57'] |
| (+)largest_factor_bitlen | 0x52 |
| (-)factorization | ['0x3', '0x1d', '0x5e293205e293205e293205e293205e28d87d38060dd8caf10b7284d9eba8acd'] |
| (-)largest_factor_bitlen | 0xfb |
| $\text{kn_factorization}(k=5)$ | |
| (+)factorization | NO DATA (timed out) |
| (+)largest_factor_bitlen | NO DATA (timed out) |
| (-)factorization | ['0x2', '0x7', '0x2db6db6db6db6db6db6db6db6db6db6d8b63a65782b8f719066a2b5af0fb519d'] |
| (-)largest_factor_bitlen | 0xfe |
| $\text{kn_factorization}(k=6)$ | |
| (+)factorization | ['0xbe9', '0x1f1b', '0xaec83ae65', '0x38ef5c968c2a40dfa5', '0xda706c088d06f5b310c137538368c7f9'] |
| (+)largest_factor_bitlen | 0x80 |
| (-)factorization | ['0x5ad', '0x679', '0x139d', '0x5873', '0x1f2e5', '0xbd28f', '0x5d2e13', '0x4cadca8b5a56e698f', '0x13a683b287832701f2db55'] |
| (-)largest_factor_bitlen | 0x55 |
| $\text{kn_factorization}(k=7)$ | |
| (+)factorization | ['0x2', '0x5', '0x29', '0x19e0d', '0x239a5', '0x4207db', '0x25aabe8a7697ecefba5dd7193d26ac0bd87b97c18da3d9cc9'] |
| (+)largest_factor_bitlen | 0xc2 |
| (-)factorization | ['0x2', '0x2', '0x3', '0xb', '0x1505', '0x18206ac3b', '0x65f326ff1', '0x89aa97a04383623b30c431edcfae55041932eb0361d'] |
| (-)largest_factor_bitlen | 0xac |
| $\text{kn_factorization}(k=8)$ | |
| (+)factorization | ['0x3', '0x3', '0x3', '0x3', '0x71', '0x155f', '0xe0bb9d', '0x281875f129c6d', '0x9bf35629b549624cbaadbf4d9527be9154d02cdf'] |
| (+)largest_factor_bitlen | 0xa0 |
| (-)factorization | ['0x5', '0x3d', '0x288d9f8af', '0x6a5a66563575', '0x57a50dcd1f23f', '0x9502d6f1a5a5b042f0f25def97665b'] |
| (-)largest_factor_bitlen | 0x78 |
| $\text{torsion_extension}(l=2)$ | |
| least | 0x3 |
| full | 0x3 |
| relative | 0x1 |
| $\text{torsion_extension}(l=3)$ | |
| least | 0x8 |
| full | 0x8 |
| relative | 0x1 |
| $\text{torsion_extension}(l=5)$ | |
| least | 0x18 |
| full | 0x18 |
| relative | 0x1 |
| $\text{torsion_extension}(l=7)$ | |
| least | 0x10 |
| full | 0x10 |
| relative | 0x1 |
| $\text{torsion_extension}(l=11)$ | |
| least | 0x5 |
| full | 0x5 |
| relative | 0x1 |
| $\text{torsion_extension}(l=13)$ | |
| least | 0xc |
| full | 0xd |
| relative | 0x1 |
| $\text{torsion_extension}(l=17)$ | |
| least | 0x10 |
| full | 0x11 |
| relative | 0x1 |
| $\text{conductor}(deg=2)$ | |
| ratio_sqrt | 0x79b5c7d7c52d4c2054705367c3a6b219 |
| factorization | ['0x47', '0x1b6d78c0d981319b1dd86954ade4a9f'] |
| $\text{conductor}(deg=3)$ | |
| ratio_sqrt | 0x4622a801d47d31929427898fe6049161ccb6472bd76d9816a4f91a0475ad3692 |
| factorization | ['0x2', '0xfe7a9b03e105c71', '0x2346fcc9815477ceb8eebb005988a9faf72ef0cebb7a9f859'] |
| $\text{conductor}(deg=4)$ | |
| ratio_sqrt | 0x5e331295970aa8f55bbd8e8d531b01ab9143f70c02378162cb9c85ecb526a74f655cbddef32d3a898a8af0cedd65a38d |
| factorization | NO DATA (timed out) |
| $\text{embedding}()$ | |
| embedding_degree_complement | 0x2 |
| complement_bit_length | 0x2 |
| $\text{class_number}()$ | |
| upper | NO DATA (timed out) |
| lower | NO DATA (timed out) |
| $\text{small_prime_order}(l=2)$ | |
| order | 0x7fffffffffffffffffffffffffffffff864a38283ad2b3dfab8fac983c594aea |
| complement_bit_length | 0x1 |
| $\text{small_prime_order}(l=3)$ | |
| order | 0xe38e38e38e38e38e38e38e38e38e38e2b5d94763f6cbea7130ff6bb94ed7a1a |
| complement_bit_length | 0x4 |
| $\text{small_prime_order}(l=5)$ | |
| order | 0x2aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa8218bd62be463bf5392fe432bec86e4e |
| complement_bit_length | 0x2 |
| $\text{small_prime_order}(l=7)$ | |
| order | 0x3fffffffffffffffffffffffffffffffc3251c141d6959efd5c7d64c1e2ca575 |
| complement_bit_length | 0x2 |
| $\text{small_prime_order}(l=11)$ | |
| order | 0x3fffffffffffffffffffffffffffffffc3251c141d6959efd5c7d64c1e2ca575 |
| complement_bit_length | 0x2 |
| $\text{small_prime_order}(l=13)$ | |
| order | 0x3fffffffffffffffffffffffffffffffc3251c141d6959efd5c7d64c1e2ca575 |
| complement_bit_length | 0x2 |
| $\text{division_polynomials}(l=2)$ | |
| factorization | [['0x3', '0x1']] |
| len | 0x1 |
| $\text{division_polynomials}(l=3)$ | |
| factorization | [['0x4', '0x1']] |
| len | 0x1 |
| $\text{division_polynomials}(l=5)$ | |
| factorization | [['0xc', '0x1']] |
| len | 0x1 |
| $\text{volcano}(l=2)$ | |
| crater_degree | 0x0 |
| depth | 0x0 |
| $\text{volcano}(l=3)$ | |
| crater_degree | 0x0 |
| depth | 0x0 |
| $\text{volcano}(l=5)$ | |
| crater_degree | 0x0 |
| depth | 0x0 |
| $\text{volcano}(l=7)$ | |
| crater_degree | 0x0 |
| depth | 0x0 |
| $\text{volcano}(l=11)$ | |
| crater_degree | 0x2 |
| depth | 0x0 |
| $\text{volcano}(l=13)$ | |
| crater_degree | 0x1 |
| depth | 0x0 |
| $\text{volcano}(l=17)$ | |
| crater_degree | 0x1 |
| depth | 0x0 |
| $\text{volcano}(l=19)$ | |
| crater_degree | 0x0 |
| depth | 0x0 |
| $\text{isogeny_extension}(l=2)$ | |
| least | 0x3 |
| full | 0x3 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=3)$ | |
| least | 0x4 |
| full | 0x4 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=5)$ | |
| least | 0x6 |
| full | 0x6 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=7)$ | |
| least | 0x8 |
| full | 0x8 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=11)$ | |
| least | 0x1 |
| full | 0x5 |
| relative | 0x5 |
| $\text{isogeny_extension}(l=13)$ | |
| least | 0x1 |
| full | 0xd |
| relative | 0xd |
| $\text{isogeny_extension}(l=17)$ | |
| least | 0x1 |
| full | 0x11 |
| relative | 0x11 |
| $\text{isogeny_extension}(l=19)$ | |
| least | 0x14 |
| full | 0x14 |
| relative | 0x1 |
| $\text{trace_factorization}(deg=1)$ | |
| trace | 0x79b5c7d7c52d4c2054705367c3a6b219 |
| trace_factorization | ['0x47', '0x1b6d78c0d981319b1dd86954ade4a9f'] |
| number_of_factors | 0x2 |
| $\text{trace_factorization}(deg=2)$ | |
| trace | 0x79b5c7d7c52d4c2054705367c3a6b219 |
| trace_factorization | ['0x3', '0x907', '0xbe9', '0xf425', '0x15175', '0x1cd34d', '0x15836857fb5858b', '0x33a3811eff119fc3d8bfb5b494508bf'] |
| number_of_factors | 0x8 |
| $\text{isogeny_neighbors}(l=2)$ | |
| len | 0x0 |
| $\text{isogeny_neighbors}(l=3)$ | |
| len | 0x0 |
| $\text{isogeny_neighbors}(l=5)$ | |
| len | 0x0 |
| $\text{q_torsion}()$ | |
| Q_torsion | 0x1 |
| $\text{hamming_x}(weight=1)$ | |
| x_coord_count | 0x75 |
| expected | 0x7f |
| ratio | 1.08547 |
| $\text{hamming_x}(weight=2)$ | |
| x_coord_count | 0x3fe9 |
| expected | 0x3f40 |
| ratio | 0.98967 |
| $\text{hamming_x}(weight=3)$ | |
| x_coord_count | 0x14dac3 |
| expected | 0x14d63f |
| ratio | 0.99915 |
| $\text{square_4p1}()$ | |
| p | 0x1 |
| order | 0x1 |
| $\text{pow_distance}()$ | |
| distance | 0x79b5c7d7c52d4c2054705367c3a6b515 |
| ratio | 3.5786799867032686e+38 |
| distance 32 | 0xb |
| distance 64 | 0x15 |
| $\text{multiples_x}(k=1)$ | |
| Hx | None |
| bits | None |
| difference | None |
| ratio | None |
| $\text{multiples_x}(k=2)$ | |
| Hx | None |
| bits | None |
| difference | None |
| ratio | None |
| $\text{multiples_x}(k=3)$ | |
| Hx | None |
| bits | None |
| difference | None |
| ratio | None |
| $\text{multiples_x}(k=4)$ | |
| Hx | None |
| bits | None |
| difference | None |
| ratio | None |
| $\text{multiples_x}(k=5)$ | |
| Hx | None |
| bits | None |
| difference | None |
| ratio | None |
| $\text{multiples_x}(k=6)$ | |
| Hx | None |
| bits | None |
| difference | None |
| ratio | None |
| $\text{multiples_x}(k=7)$ | |
| Hx | None |
| bits | None |
| difference | None |
| ratio | None |
| $\text{multiples_x}(k=8)$ | |
| Hx | None |
| bits | None |
| difference | None |
| ratio | None |
| $\text{multiples_x}(k=9)$ | |
| Hx | None |
| bits | None |
| difference | None |
| ratio | None |
| $\text{multiples_x}(k=10)$ | |
| Hx | None |
| bits | None |
| difference | None |
| ratio | None |
| $\text{x962_invariant}()$ | |
| r | 0x260ad32bdc94c614c23b58d6abadef723c97eef8e0f7d98557a50870710bbd8b |
| $\text{brainpool_overlap}()$ | |
| o | 0x3ffffffffffffffffffffd01 |
| $\text{weierstrass}()$ | |
| a | 0x7ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffd00 |
| b | -0x51bd |